Introduction

Relevance of the Topic

Matrices, as fundamental tools in the theory and application of Linear Systems, represent a natural extension of knowledge acquired in previous years of mathematics.

Linear Systems themselves are one of the most ubiquitous mathematical structures found in science and engineering, with applications ranging from solving simple equation problems to modeling complex phenomena.

In this context, the transition to matrix notation for representing Linear Systems provides a more compact and efficient way of working with these systems, preparing the student for more advanced studies in calculus and linear algebra.

Contextualization

Within the High School mathematics curriculum, the study of Linear Systems falls under the subarea of algebra. This is a crucial topic that provides the foundation for more advanced concepts, such as vector spaces, linear applications, and linear transformations.

Understanding Linear Systems and, more specifically, the ability to write them in matrix form, is an essential prerequisite for many fields of study and professional careers, including exact sciences, engineering, computer science, economics, statistics, and many others.

Therefore, the study of the theme 'Linear Systems: Written by Matrices' is perfectly situated within the context of preparing students for future studies and providing them with the necessary tools for a variety of academic and professional applications.

Theoretical Development

Components

• Linear System: A linear system is a set of one or more linear equations, with the same set of variables. It is a central area of linear algebra, as it forms the basis for many other concepts in this field.

• Linear Equations: Individually, each equation is just a mathematical sentence that can be true or false. In the context of a linear system, however, the equations interact with each other, forming a complex system of linear relationships.

• Matrices: A matrix is a rectangular arrangement of numbers, symbols, or expressions, organized in rows and columns. Matrices are especially relevant in the context of linear systems, as the coefficients of the equations and the constants can be organized into matrices to make the representation more concise and manipulable.

• Matrix Notation: Matrix notation for linear systems is a convenient way to represent these systems of equations, transforming the unknowns and coefficients into matrices and vectors. This notation simplifies the process of solving linear systems, especially when dealing with systems with many equations and variables.

Key Terms

• Augmented Matrix: An augmented matrix is the combination of two matrices - the matrix of coefficients and the matrix of independent terms - to form a single matrix. It is used to represent the matrix notation of linear systems.

• Coefficient Matrix: A matrix that contains the coefficients of the variables in a linear system.

• Identity Matrix: A square matrix in which all elements off the main diagonal are zero, and all elements on the main diagonal are one.

Examples and Cases

• Example of Linear System and Augmented Matrix:

    Linear System:
    2x + y = 5
    3x - 2y = -8
    
    Augmented Matrix:
    | 2  1 |  5 |
    | 3 -2 | -8 |
    ```

In this example, the linear system is represented by a 2x2 augmented matrix. The first column contains the coefficients of x, the second column contains the coefficients of y, and the third column contains the independent terms.

- **Example of Solving Linear System Written by Matrix**:

Linear System:
2x + y = 5
3x - 2y = -8

Augmented Matrix:
| 2  1 |  5 |
| 3 -2 | -8 |

We can perform the row operation:

| 1  0 | 2 |
| 0  1 | 1 |

Therefore, the solution to the linear system is x = 2 and y = 1.

In this example, the linear system is solved through the row operation of the augmented matrix. The resulting matrix after the row operation represents the solution to the linear system, with the first column representing x and the second column representing y.

# **Detailed Summary**

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## **Key Points**

- **Understanding of Linear System**: The ability to understand what a linear system is - a set of linear equations with the same set of variables - is fundamental. A linear system can have one, no, or infinite solutions.

- **Utility of Matrices**: Matrices are an essential tool for the representation and solution of linear systems. The coefficient matrix and the matrix of independent terms can be combined into an augmented matrix for a concise and efficient representation of the linear system.

- **Importance of Matrix Notation**: Matrix notation is a powerful way to represent linear systems, making problem-solving simpler and more direct. The use of the augmented matrix, which combines the coefficient matrix and the matrix of independent terms, is especially useful in this context.

- **Row Operation**: The row operation of an augmented matrix is a key tool in solving linear systems. This operation consists of a series of transformations that result in a matrix with a solution that is easier to identify.

- **Importance of Practice**: Solving linear systems written in matrix notation requires practice. It is important to become familiar with the notation, understand how to manipulate matrices, and practice the row operation.

## **Conclusions**

- **Space Saving and Efficiency**: The transition to matrix notation for representing linear systems offers a more compact and efficient way of working with these systems. With matrix notation, all equations of a system can be represented in a single augmented matrix.

- **Ease of Solution**: Matrix notation, along with row operation, makes solving linear systems simpler and more direct. With practice, this is a tool that students can master and effectively use in their future studies and professional careers.

## **Suggested Exercises**

1. **Exercise of Translating Linear System to Matrix Notation**: Given the linear system `4x + 3y = 9` and `2x - y = 4`, write it in matrix notation.

2. **Row Operation Exercise**: Given the augmented matrix `| 1 2 | 3  6 |`, perform the row operation and indicate the solution of the corresponding linear system.

3. **Exercise of Solving Linear System by Augmented Matrix**: Given the augmented matrix `| 2  1 |  5 |`, solve the corresponding linear system and compare it with the original solution.